Page 316 - Elements of Distribution Theory

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302 Orthogonal Polynomials

Example 10.1 (Legendre polynomials). Let F denote the distribution function of the uni-
form distribution on (−1, 1); the orthogonal polynomials with respect to this distribution
are known as the Legendre polynomials. They are traditionally denoted by P 0 , P 1 ,... and
we will use that notation here.
Note that

∞ 1
1 1
n n n+1 if n is even
x dF(x) = x dx = .
2
−∞ −1 0 if n is odd
Hence, using the procedure described in Theorem 10.2, we have that P 0 (x) = 1,

x 1
P 1 (x) = det = x
01
and
 2 
x x 0
1 2 1
P 2 (x) = det  1/3 0 1  =− x + .
3 9
0 1/30
It is easy to verify directly that these polynomials are orthogonal with respect to F.

If p 0 , p 1 ,... are orthogonal polynomials with respect to some distribution function
F, then so are α 0 p 0 ,α 1 p 1 ,... for any nonzero constants α 0 ,α 1 ,.... Hence, orthogonal
polynomials are generally standardized in some way. Typically, one of the following stan-
n
dardizations is used: the coefficient of x in p n (x)is required to be 1, it is required that
p n (1) = 1, or p n must satisfy
∞

2
p n (x) dF(x) = 1. (10.3)
−∞
Example 10.2 (Legendre polynomials). Consider the Legendre polynomials described in
Example 10.1. If we require that the polynomials have lead coefficient 1, then
1
2
P 0 (x) = 1, P 1 (x) = x, and P 2 (x) = x − .
3
If we require that p n (1) = 1,
3 2 1
P 0 (x) = 1, P 1 (x) = x, and P 2 (x) = x − .
2 2
If we require that (10.3) holds, then
√
√ √ 3 2 5
P 0 (x) = 1, P 1 (x) = 3x, and P 2 (x) = 5 x − .
2 2
For the Legendre polynomials, the second of these standardizations is commonly used and
that is the one we will use here.

The following result gives another approach to finding orthogonal polynomials.

Theorem 10.3. Let F denote the distribution function of an absolutely continuous distri-
bution with support [a, b], −∞ ≤ a < b ≤∞, and let p denote the corresponding density